civil engineering

2. If i, j, k are orthonormal vectors and A = Axi + Ay j + Azk then |A|2
= A2
x + A2
y + A2
z. [Orthonormal vectors ≡
orthogonal unit vectors.]
Scalar product
A · B = |A| |B| cosθ where θ is the angle between the vectors
= AxBx + AyBy + AzBz = [ Ax Ay Az ]


Bx
By
Bz


Scalar multiplication is commutative: A · B = B · A.
Equation of a line
A point r ≡ (x, y, z) lies on a line passing through a point a and parallel to vector b if
r = a + λb
with λ a real number.
Vector Algebra
Page 1 of 24
ENGINEERING
MATHEMATICS FORMULAS &
SHORT NOTES HANDBOOK

3. Equation of a plane
A point r ≡ (x, y, z) is on a plane if either
(a) r · b
d = |d|, where d is the normal from the origin to the plane, or
(b)
x
X
+
y
Y
+
z
Z
= 1 where X, Y, Z are the intercepts on the axes.
Vector product
A×B = n |A| |B| sinθ, whereθ is the angle between the vectors and n is a unit vector normal to the plane containing
A and B in the direction for which A, B, n form a right-handed set of axes.
A × B in determinant form
i j k
Ax Ay Az
Bx By Bz
A × B in matrix form


0 −Az Ay
Az 0 −Ax
−Ay Ax 0




Bx
By
Bz


Vector multiplication is not commutative: A × B = −B × A.
Scalar triple product
A × B · C = A · B × C =
Ax Ay Az
Bx By Bz
Cx Cy Cz
= −A × C · B, etc.
Vector triple product
A × (B × C) = (A · C)B − (A · B)C, (A × B) × C = (A · C)B − (B · C)A
Non-orthogonal basis
A = A1e1 + A2e2 + A3e3
A1 = 0
· A where 0
=
e2 × e3
e1 · (e2 × e3)
Similarly for A2 and A3.
Summation convention
a = aiei implies summation over i = 1 . . . 3
a · b = aibi
(a × b)i = εijkajbk where ε123 = 1; εijk = −εikj
εijkεklm = δilδjm − δimδjl
Page 2 of 24

4. Unit matrices
The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements
zero, i.e., (I)ij = δij. If A is a square matrix of order n, then AI = IA = A. Also I = I−1
.
I is sometimes written as In if the order needs to be stated explicitly.
Products
If A is a (n × l) matrix and B is a (l × m) then the product AB is defined by
(AB)ij =
l
∑
k=1
AikBkj
In general AB 6= BA.
Transpose matrices
If A is a matrix, then transpose matrix AT
is such that (AT
)ij = (A)ji.
Inverse matrices
If A is a square matrix with non-zero determinant, then its inverse A−1
is such that AA−1
= A−1
A = I.
(A−1
)ij =
transpose of cofactor of Aij
|A|
where the cofactor of Aij is (−1)i+j
times the determinant of the matrix A with the j-th row and i-th column deleted.
Determinants
If A is a square matrix then the determinant of A, |A| (≡ det A) is defined by
|A| = ∑
i,j,k,...
ijk...A1i A2j A3k . . .
where the number of the suffixes is equal to the order of the matrix.
2×2 matrices
If A =
a b
c d
then,
|A| = ad − bc AT
=
a c
b d
A−1
=
1
|A|
d −b
−c a
Product rules
(AB . . . N)T
= NT
. . . BT
AT
(AB . . . N)−1
= N−1
. . . B−1
A−1
(if individual inverses exist)
|AB . . . N| = |A| |B| . . . |N| (if individual matrices are square)
Orthogonal matrices
An orthogonal matrix Q is a square matrix whose columns qi form a set of orthonormal vectors. For any orthogonal
matrix Q,
Q−1
= QT
, |Q| = ±1, QT
is also orthogonal.
Matrix Algebra
Page 3 of 24

5. Solving sets of linear simultaneous equations
If A is square then Ax = b has a unique solution x = A−1
b if A−1
exists, i.e., if |A| 6= 0.
If A is square then Ax = 0 has a non-trivial solution if and only if |A| = 0.
An over-constrained set of equations Ax = b is one in which A has m rows and n columns, where m (the number
of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the
error |Ax − b|) is the solution of the n equations AT
Ax = AT
b. If the columns of A are orthonormal vectors then
x = AT
b.
Hermitian matrices
The Hermitian conjugate of A is A† = (A∗
)T
, where A∗
is a matrix each of whose components is the complex
conjugate of the corresponding components of A. If A = A† then A is called a Hermitian matrix.
Eigenvalues and eigenvectors
The n eigenvalues λi and eigenvectors ui of an n × n matrix A are the solutions of the equation Au = λu. The
eigenvalues are the zeros of the polynomial of degree n, Pn(λ) = |A − λI|. If A is Hermitian then the eigenvalues
λi are real and the eigenvectors ui are mutually orthogonal. |A − λI| = 0 is called the characteristic equation of the
matrix A.
Tr A = ∑
i
λi, also |A| = ∏
i
λi.
If S is a symmetric matrix, Λ is the diagonal matrix whose diagonal elements are the eigenvalues of S, and U is the
matrix whose columns are the normalized eigenvectors of A, then
UT
SU = Λ and S = UΛUT
.
If x is an approximation to an eigenvector of A then xT
Ax/(xT
x) (Rayleigh’s quotient) is an approximation to the
corresponding eigenvalue.
Commutators
[A, B] ≡ AB − BA
[A, B] = −[B, A]
[A, B]† = [B†, A†]
[A + B, C] = [A, C] + [B, C]
[AB, C] = A[B, C] + [A, C]B
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
Hermitian algebra
b† = (b∗
1, b∗
2, . . .)
Matrix form Operator form Bra-ket form
Hermiticity b∗
· A · c = (A · b)∗
· c
Z
ψ∗
Oφ =
Z
(Oψ)∗
φ hψ|O|φi
Eigenvalues, λ real Aui = λ(i)ui Oψi = λ(i)ψi O |ii = λi |ii
Orthogonality ui · uj = 0
Z
ψ∗
i ψj = 0 hi|ji = 0 (i 6= j)
Completeness b = ∑
i
ui(ui · b) φ = ∑
i
ψi
Z
ψ∗
i φ
φ = ∑
i
|ii hi|φi
Rayleigh–Ritz
Lowest eigenvalue λ0 ≤
b∗
· A · b
b∗
· b
λ0 ≤
Z
ψ∗
Oψ
Z
ψ∗
ψ
hψ|O|ψi
hψ|ψi
Page 4 of 24

6. Pauli spin matrices
σx =
0 1
1 0
, σy =
0 −i
i 0
, σz =
1 0
0 −1
σxσy = iσz, σyσz = iσx, σzσx = iσy, σxσx = σyσy = σzσz = I
Notation
φ is a scalar function of a set of position coordinates. In Cartesian coordinates
φ = φ(x, y, z); in cylindrical polar coordinates φ = φ(ρ,ϕ, z); in spherical
polar coordinates φ = φ(r,θ,ϕ); in cases with radial symmetry φ = φ(r).
A is a vector function whose components are scalar functions of the position
coordinates: in Cartesian coordinates A = iAx + jAy + kAz, where Ax, Ay, Az
are independent functions of x, y, z.
In Cartesian coordinates ∇ (‘del’) ≡ i
∂
∂x
+ j
∂
∂y
+ k
∂
∂z
≡









∂
∂x
∂
∂y
∂
∂z









gradφ = ∇φ, div A = ∇ · A, curl A = ∇ × A
Identities
grad(φ1 + φ2) ≡ gradφ1 + gradφ2 div(A1 + A2) ≡ div A1 + div A2
grad(φ1φ2) ≡ φ1 gradφ2 + φ2 gradφ1
curl(A + A) ≡ curl A1 + curl A2
div(φA) ≡ φ div A + (gradφ) · A, curl(φA) ≡ φ curl A + (gradφ) × A
div(A1 × A2) ≡ A2 · curl A1 − A1 · curl A2
curl(A1 × A2) ≡ A1 div A2 − A2 div A1 + (A2 · grad)A1 − (A1 · grad)A2
div(curl A) ≡ 0, curl(gradφ) ≡ 0
curl(curl A) ≡ grad(div A) − div(grad A) ≡ grad(div A) − ∇2
A
grad(A1 · A2) ≡ A1 × (curl A2) + (A1 · grad)A2 + A2 × (curl A1) + (A2 · grad)A1
Vector Calculus
Page 5 of 24

7. Grad, Div, Curl and the Laplacian
Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates
Conversion to
Cartesian
Coordinates
x = ρ cosϕ y = ρ sinϕ z = z
x = r cosϕ sinθ y = r sinϕ sinθ
z = r cosθ
Vector A Axi + Ay j + Azk Aρb
ρ + Aϕ b
ϕ + Azb
z Arb
r + Aθ
b
θ + Aϕ b
ϕ
Gradient ∇φ
∂φ
∂x
i +
∂φ
∂y
j +
∂φ
∂z
k
∂φ
∂ρ
b
ρ +
1
ρ
∂φ
∂ϕ
b
ϕ +
∂φ
∂z
b
z
∂φ
∂r
b
r +
1
r
∂φ
∂θ
b
θ +
1
r sinθ
∂φ
∂ϕ
b
ϕ
Divergence
∇ · A
∂Ax
∂x
+
∂Ay
∂y
+
∂Az
∂z
1
ρ
∂(ρAρ)
∂ρ
+
1
ρ
∂Aϕ
∂ϕ
+
∂Az
∂z
1
r2
∂(r2
Ar)
∂r
+
1
r sinθ
∂Aθ sinθ
∂θ
+
1
r sinθ
∂Aϕ
∂ϕ
Curl ∇ × A
i j k
∂
∂x
∂
∂y
∂
∂z
Ax Ay Az
1
ρ
b
ρ b
ϕ
1
ρ
b
z
∂
∂ρ
∂
∂ϕ
∂
∂z
Aρ ρAϕ Az
1
r2
sinθ
b
r
1
r sinθ
b
θ
1
r
b
ϕ
∂
∂r
∂
∂θ
∂
∂ϕ
Ar rAθ rAϕ sinθ
Laplacian
∇2
φ
∂2
φ
∂x2
+
∂2
φ
∂y2
+
∂2
φ
∂z2
1
ρ
∂
∂ρ
ρ
∂φ
∂ρ
+
1
ρ2
∂2
φ
∂ϕ2
+
∂2
φ
∂z2
1
r2
∂
∂r
r2 ∂φ
∂r
+
1
r2
sinθ
∂
∂θ
sinθ
∂φ
∂θ
+
1
r2
sin2
θ
∂2
φ
∂ϕ2
Transformation of integrals
L = the distance along some curve ‘C’ in space and is measured from some fixed point.
S = a surface area
τ = a volume contained by a specified surface
b
t = the unit tangent to C at the point P
b
n = the unit outward pointing normal
A = some vector function
dL = the vector element of curve (= b
t dL)
dS = the vector element of surface (= b
n dS)
Then
Z
C
A · b
t dL =
Z
C
A · dL
and when A = ∇φ
Z
C
(∇φ) · dL =
Z
C
dφ
Gauss’s Theorem (Divergence Theorem)
When S defines a closed region having a volume τ
Z
τ
(∇ · A) dτ =
Z
S
(A · b
n) dS =
Z
S
A · dS
also
Z
τ
(∇φ) dτ =
Z
S
φ dS
Z
τ
(∇ × A) dτ =
Z
S
(b
n × A) dS
Page 6 of 24

8. Stokes’s Theorem
When C is closed and bounds the open surface S,
Z
S
(∇ × A) · dS =
Z
C
A · dL
also
Z
S
(b
n × ∇φ) dS =
Z
C
φ dL
Green’s Theorem
Z
S
ψ∇φ · dS =
Z
τ
∇ · (ψ∇φ) dτ
=
Z
τ
ψ∇2
φ + (∇ψ) · (∇φ)
dτ
Green’s Second Theorem
Z
τ
(ψ∇2
φ − φ∇2
ψ) dτ =
Z
S
[ψ(∇φ) − φ(∇ψ)] · dS
Complex numbers
The complex number z = x + iy = r(cosθ + i sinθ) = r ei(θ+2nπ)
, where i2
= −1 and n is an arbitrary integer. The
real quantity r is the modulus of z and the angle θ is the argument of z. The complex conjugate of z is z∗
= x − iy =
r(cosθ − i sinθ) = r e−iθ
; zz∗
= |z|2
= x2
+ y2
De Moivre’s theorem
(cosθ + i sinθ)n
= einθ
= cos nθ + i sin nθ
Power series for complex variables.
ez
= 1 + z +
z2
2!
+ · · · +
zn
n!
+ · · · convergent for all finite z
sin z = z −
z3
3!
+
z5
5!
− · · · convergent for all finite z
cos z = 1 −
z2
2!
+
z4
4!
− · · · convergent for all finite z
ln(1 + z) = z −
z2
2
+
z3
3
− · · · principal value of ln(1 + z)
This last series converges both on and within the circle |z| = 1 except at the point z = −1.
tan−1
z = z −
z3
3
+
z5
5
− · · ·
This last series converges both on and within the circle |z| = 1 except at the points z = ±i.
(1 + z)n
= 1 + nz +
n(n − 1)
2!
z2
+
n(n − 1)(n − 2)
3!
z3
+ · · ·
This last series converges both on and within the circle |z| = 1 except at the point z = −1.
Page 7 of 24
Complex Variables

9. cos2
A + sin2
A = 1 sec2
A − tan2
A = 1 cosec2
A − cot2
A = 1
sin 2A = 2 sin A cos A cos 2A = cos2
A − sin2
A tan 2A =
2 tan A
1 − tan2
A
.
sin(A ± B) = sin A cos B ± cos A sin B cos A cos B =
cos(A + B) + cos(A − B)
2
cos(A ± B) = cos A cos B ∓ sin A sin B sin A sin B =
cos(A − B) − cos(A + B)
2
tan(A ± B) =
tan A ± tan B
1 ∓ tan A tan B
sin A cos B =
sin(A + B) + sin(A − B)
2
sin A + sin B = 2 sin
A + B
2
cos
A − B
2
sin A − sin B = 2 cos
A + B
2
sin
A − B
2
cos A + cos B = 2 cos
A + B
2
cos
A − B
2
cos A − cos B = −2 sin
A + B
2
sin
A − B
2
cos2
A =
1 + cos 2A
2
sin2
A =
1 − cos 2A
2
cos3
A =
3 cos A + cos 3A
4
sin3
A =
3 sin A − sin 3A
4
Relations between sides and angles of any plane triangle
In a plane triangle with angles A, B, and C and sides opposite a, b, and c respectively,
a
sin A
=
b
sin B
=
c
sin C
= diameter of circumscribed circle.
a2
= b2
+ c2
− 2bc cos A
a = b cos C + c cos B
cos A =
b2
+ c2
− a2
2bc
tan
A − B
2
=
a − b
a + b
cot
C
2
area =
1
2
ab sin C =
1
2
bc sin A =
1
2
ca sin B =
q
s(s − a)(s − b)(s − c), where s =
1
2
(a + b + c)
Relations between sides and angles of any spherical triangle
In a spherical triangle with angles A, B, and C and sides opposite a, b, and c respectively,
sin a
sin A
=
sin b
sin B
=
sin c
sin C
cos a = cos b cos c + sin b sin c cos A
cos A = − cos B cos C + sin B sin C cos a
Page 8 of 24
Trigonometric Formulae

10. cosh x =
1
2
( ex
+ e−x
) = 1 +
x2
2!
+
x4
4!
+ · · · valid for all x
sinh x =
1
2
( ex
− e−x
) = x +
x3
3!
+
x5
5!
+ · · · valid for all x
cosh ix = cos x cos ix = cosh x
sinh ix = i sin x sin ix = i sinh x
tanh x =
sinh x
cosh x
sech x =
1
cosh x
coth x =
cosh x
sinh x
cosech x =
1
sinh x
cosh2
x − sinh2
x = 1
For large positive x:
cosh x ≈ sinh x →
ex
2
tanh x → 1
For large negative x:
cosh x ≈ − sinh x →
e−x
2
tanh x → −1
Relations of the functions
sinh x = − sinh(−x) sech x = sech(−x)
cosh x = cosh(−x) cosech x = − cosech(−x)
tanh x = − tanh(−x) coth x = − coth(−x)
sinh x =
2 tanh (x/2)
1 − tanh2
(x/2)
=
tanh x
q
1 − tanh2
x
cosh x =
1 + tanh2
(x/2)
1 − tanh2
(x/2)
=
1
q
1 − tanh2
x
tanh x =
q
1 − sech2
x sech x =
q
1 − tanh2
x
coth x =
q
cosech2
x + 1 cosech x =
q
coth2
x − 1
sinh(x/2) =
r
cosh x − 1
2
cosh(x/2) =
r
cosh x + 1
2
tanh(x/2) =
cosh x − 1
sinh x
=
sinh x
cosh x + 1
sinh(2x) = 2 sinh x cosh x tanh(2x) =
2 tanh x
1 + tanh2
x
cosh(2x) = cosh2
x + sinh2
x = 2 cosh2
x − 1 = 1 + 2 sinh2
x
sinh(3x) = 3 sinh x + 4 sinh3
x cosh 3x = 4 cosh3
x − 3 cosh x
tanh(3x) =
3 tanh x + tanh3
x
1 + 3 tanh2
x
Page 9 of 24
Hyperbolic Functions

11. sinh(x ± y) = sinh x cosh y ± cosh x sinh y
cosh(x ± y) = cosh x cosh y ± sinh x sinh y
tanh(x ± y) =
tanh x ± tanh y
1 ± tanh x tanh y
sinh x + sinh y = 2 sinh
1
2
(x + y) cosh
1
2
(x − y) cosh x + cosh y = 2 cosh
1
2
(x + y) cosh
1
2
(x − y)
sinh x − sinh y = 2 cosh
1
2
(x + y) sinh
1
2
(x − y) cosh x − cosh y = 2 sinh
1
2
(x + y) sinh
1
2
(x − y)
sinh x ± cosh x =
1 ± tanh (x/2)
1 ∓ tanh(x/2)
= e±x
tanh x ± tanh y =
sinh(x ± y)
cosh x cosh y
coth x ± coth y = ±
sinh(x ± y)
sinh x sinh y
Inverse functions
sinh−1 x
a
= ln
x +
p
x2 + a2
a
!
for −∞ x ∞
cosh−1 x
a
= ln
x +
p
x2 − a2
a
!
for x ≥ a
tanh−1 x
a
=
1
2
ln
a + x
a − x
for x2
a2
coth−1 x
a
=
1
2
ln
x + a
x − a
for x2
a2
sech−1 x
a
= ln

 a
x
+
s
a2
x2
− 1

 for 0 x ≤ a
cosech−1 x
a
= ln

 a
x
+
s
a2
x2
+ 1

 for x 6= 0
nc
xn
→ 0 as n → ∞ if |x| 1 (any fixed c)
xn
/n! → 0 as n → ∞ (any fixed x)
(1 + x/n)n
→ ex
as n → ∞, x ln x → 0 as x → 0
If f (a) = g(a) = 0 then lim
x→a
f (x)
g(x)
=
f 0
(a)
g0
(a)
(l’Hôpital’s rule)
12
Page 10 of 24
Limits

12. (uv)0
= u0
v + uv0
,
u
v
0
=
u0
v − uv0
v2
(uv)(n)
= u(n)
v + nu(n−1)
v(1)
+ · · · + n
Cru(n−r)
v(r)
+ · · · + uv(n)
Leibniz Theorem
where n
Cr ≡
n
r
=
n!
r!(n − r)!
d
dx
(sin x) = cos x
d
dx
(sinh x) = cosh x
d
dx
(cos x) = − sin x
d
dx
(cosh x) = sinh x
d
dx
(tan x) = sec2
x
d
dx
(tanh x) = sech2
x
d
dx
(sec x) = sec x tan x
d
dx
(sech x) = − sech x tanh x
d
dx
(cot x) = − cosec2
x
d
dx
(coth x) = − cosech2
x
d
dx
(cosec x) = − cosec x cot x
d
dx
(cosech x) = − cosech x coth x
Standard forms
Z
xn
dx =
xn+1
n + 1
+ c for n 6= −1
Z
1
x
dx = ln x + c
Z
ln x dx = x(ln x − 1) + c
Z
eax
dx =
1
a
eax
+ c
Z
x eax
dx = eax
x
a
−
1
a2
+ c
Z
x ln x dx =
x2
2
ln x −
1
2
+ c
Z
1
a2
+ x2
dx =
1
a
tan−1
x
a
+ c
Z
1
a2
− x2
dx =
1
a
tanh−1
x
a
+ c =
1
2a
ln
a + x
a − x
+ c for x2
a2
Z
1
x2
− a2
dx = −
1
a
coth−1
x
a
+ c =
1
2a
ln
x − a
x + a
+ c for x2
a2
Z
x
(x2
± a2
)n
dx =
−1
2(n − 1)
1
(x2
± a2
)n−1
+ c for n 6= 1
Z
x
x2
± a2
dx =
1
2
ln(x2
± a2
) + c
Z
1
p
a2 − x2
dx = sin−1
x
a
+ c
Z
1
p
x2 ± a2
dx = ln
x +
p
x2 ± a2
+ c
Z
x
p
x2 ± a2
dx =
p
x2 ± a2 + c
Z p
a2 − x2 dx =
1
2
h
x
p
a2 − x2 + a2
sin−1
x
a
i
+ c
Page 11 of 24
Differentiation
Integration

13. Z ∞
0
1
(1 + x)xp dx = π cosec pπ for p 1
Z ∞
0
cos(x2
) dx =
Z ∞
0
sin(x2
) dx =
1
2
r
π
2
Z ∞
−∞
exp(−x2
/2σ2
) dx = σ
√
2π
Z ∞
−∞
xn
exp(−x2
/2σ2
) dx =



1 × 3 × 5 × · · · (n − 1)σn+1
√
2π
0
for n ≥ 2 and even
for n ≥ 1 and odd
Z
sin x dx = − cos x + c
Z
sinh x dx = cosh x + c
Z
cos x dx = sin x + c
Z
cosh x dx = sinh x + c
Z
tan x dx = − ln(cos x) + c
Z
tanh x dx = ln(cosh x) + c
Z
cosec x dx = ln(cosec x − cot x) + c
Z
cosech x dx = ln [tanh(x/2)] + c
Z
sec x dx = ln(sec x + tan x) + c
Z
sech x dx = 2 tan−1
( ex
) + c
Z
cot x dx = ln(sin x) + c
Z
coth x dx = ln(sinh x) + c
Z
sin mx sin nx dx =
sin(m − n)x
2(m − n)
−
sin(m + n)x
2(m + n)
+ c if m2
6= n2
Z
cos mx cos nx dx =
sin(m − n)x
2(m − n)
+
sin(m + n)x
2(m + n)
+ c if m2
6= n2
Standard substitutions
If the integrand is a function of: substitute:
(a2
− x2
) or
p
a2 − x2 x = a sinθ or x = a cosθ
(x2
+ a2
) or
p
x2 + a2 x = a tanθ or x = a sinhθ
(x2
− a2
) or
p
x2 − a2 x = a secθ or x = a coshθ
If the integrand is a rational function of sin x or cos x or both, substitute t = tan(x/2) and use the results:
sin x =
2t
1 + t2
cos x =
1 − t2
1 + t2
dx =
2 dt
1 + t2
.
If the integrand is of the form: substitute:
Z
dx
(ax + b)
p
px + q
px + q = u2
Z
dx
(ax + b)
q
px2 + qx + r
ax + b =
1
u
.
Page 12 of 24

14. Integration by parts
Z b
a
u dv = uv
b
a
−
Z b
a
v du
Differentiation of an integral
If f (x,α) is a function of x containing a parameter α and the limits of integration a and b are functions of α then
d
dα
Z b(α)
a(α)
f (x,α) dx = f (b,α)
db
dα
− f (a,α)
da
dα
+
Z b(α)
a(α)
∂
∂α
f (x,α) dx.
Special case,
d
dx
Z x
a
f (y) dy = f (x).
Dirac δ-‘function’
δ(t − τ) =
1
2π
Z ∞
−∞
exp[iω(t − τ)] dω.
If f (t) is an arbitrary function of t then
Z ∞
−∞
δ(t − τ) f (t) dt = f (τ).
δ(t) = 0 if t 6= 0, also
Z ∞
−∞
δ(t) dt = 1
Reduction formulae
Factorials
n! = n(n − 1)(n − 2) . . . 1, 0! = 1.
Stirling’s formula for large n: ln(n!) ≈ n ln n − n.
For any p −1,
Z ∞
0
xp
e−x
dx = p
Z ∞
0
xp−1
e−x
dx = p!. (−1/2)! =
√
π, (1/2)! =
√
π/2, etc.
For any p, q −1,
Z 1
0
xp
(1 − x)q
dx =
p!q!
(p + q + 1)!
.
Trigonometrical
If m, n are integers,
Z π/2
0
sinm
θ cosn
θ dθ =
m − 1
m + n
Z π/2
0
sinm−2
θ cosn
θ dθ =
n − 1
m + n
Z π/2
0
sinm
θ cosn−2
θ dθ
and can therefore be reduced eventually to one of the following integrals
Z π/2
0
sinθ cosθ dθ =
1
2
,
Z π/2
0
sinθ dθ = 1,
Z π/2
0
cosθ dθ = 1,
Z π/2
0
dθ =
π
2
.
Other
If In =
Z ∞
0
xn
exp(−αx2
) dx then In =
(n − 1)
2α
In−2, I0 =
1
2
r
π
α
, I1 =
1
2α
.
Page 13 of 24

15. Diffusion (conduction) equation
∂ψ
∂t
= κ∇2
ψ
Wave equation
∇2
ψ =
1
c2
∂2
ψ
∂t2
Legendre’s equation
(1 − x2
)
d2
y
dx2
− 2x
dy
dx
+ l(l + 1)y = 0,
solutions of which are Legendre polynomials Pl(x), where Pl(x) =
1
2l
l!
d
dx
l
x2
− 1
l
, Rodrigues’ formula so
P0(x) = 1, P1(x) = x, P2(x) =
1
2
(3x2
− 1) etc.
Recursion relation
Pl(x) =
1
l
[(2l − 1)xPl−1(x) − (l − 1)Pl−2(x)]
Orthogonality
Z 1
−1
Pl(x)Pl0 (x) dx =
2
2l + 1
δll0
Bessel’s equation
x2 d2
y
dx2
+ x
dy
dx
+ (x2
− m2
)y = 0,
solutions of which are Bessel functions Jm(x) of order m.
Series form of Bessel functions of the first kind
Jm(x) =
∞
∑
k=0
(−1)k
(x/2)m+2k
k!(m + k)!
(integer m).
The same general form holds for non-integer m 0.
Page 14 of 24
Differential Equations

16. Laplace’s equation
∇2
u = 0
If expressed in two-dimensional polar coordinates (see section 4), a solution is
u(ρ,ϕ) =
Aρn
+ Bρ−n
C exp(inϕ) + D exp(−inϕ)
where A, B, C, D are constants and n is a real integer.
If expressed in three-dimensional polar coordinates (see section 4) a solution is
u(r,θ,ϕ) =
Arl
+ Br−(l+1)
Pm
l
C sin mϕ + D cos mϕ
where l and m are integers with l ≥ |m| ≥ 0; A, B, C, D are constants;
Pm
l (cosθ) = sin|m|
θ
d
d(cosθ)
|m|
Pl(cosθ)
is the associated Legendre polynomial.
P0
l (1) = 1.
If expressed in cylindrical polar coordinates (see section 4), a solution is
u(ρ,ϕ, z) = Jm(nρ)
A cos mϕ + B sin mϕ
C exp(nz) + D exp(−nz)
where m and n are integers; A, B, C, D are constants.
Spherical harmonics
The normalized solutions Ym
l (θ,ϕ) of the equation
1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin2
θ
∂2
∂ϕ2
Ym
l + l(l + 1)Ym
l = 0
are called spherical harmonics, and have values given by
Ym
l (θ,ϕ) =
s
2l + 1
4π
(l − |m|)!
(l + |m|)!
Pm
l (cosθ) eimϕ
×
(−1)m
for m ≥ 0
1 for m 0
i.e., Y0
0 =
r
1
4π
, Y0
1 =
r
3
4π
cosθ, Y±1
1 = ∓
r
3
8π
sinθ e±iϕ
, etc.
Orthogonality
Z
4π
Y∗m
l Ym0
l0 dΩ = δll0δmm0
The condition for I =
Z b
a
F(y, y0
, x) dx to have a stationary value is
∂F
∂y
=
d
dx
∂F
∂y0
, where y0
=
dy
dx
. This is the
Euler–Lagrange equation.
Page 15 of 24
Calculus of Variations

17. If φ = f (x, y, z, . . .) then
∂φ
∂x
implies differentiation with respect to x keeping y, z, . . . constant.
dφ =
∂φ
∂x
dx +
∂φ
∂y
dy +
∂φ
∂z
dz + · · · and δφ ≈
∂φ
∂x
δx +
∂φ
∂y
δy +
∂φ
∂z
δz + · · ·
where x, y, z, . . . are independent variables.
∂φ
∂x
is also written as
∂φ
∂x
y,...
or
∂φ
∂x y,...
when the variables kept
constant need to be stated explicitly.
If φ is a well-behaved function then
∂2
φ
∂x ∂y
=
∂2
φ
∂y ∂x
etc.
If φ = f (x, y),
∂φ
∂x
y
=
1
∂x
∂φ
y
,
∂φ
∂x
y
∂x
∂y
φ
∂y
∂φ
x
= −1.
Taylor series for two variables
If φ(x, y) is well-behaved in the vicinity of x = a, y = b then it has a Taylor series
φ(x, y) = φ(a + u, b + v) = φ(a, b) + u
∂φ
∂x
+ v
∂φ
∂y
+
1
2!
u2 ∂2
φ
∂x2
+ 2uv
∂2
φ
∂x ∂y
+ v2 ∂2
φ
∂y2
+ · · ·
where x = a + u, y = b + v and the differential coefficients are evaluated at x = a, y = b
Stationary points
A function φ = f (x, y) has a stationary point when
∂φ
∂x
=
∂φ
∂y
= 0. Unless
∂2
φ
∂x2
=
∂2
φ
∂y2
=
∂2
φ
∂x ∂y
= 0, the following
conditions determine whether it is a minimum, a maximum or a saddle point.
Minimum:
∂2
φ
∂x2
0, or
∂2
φ
∂y2
0,
Maximum:
∂2
φ
∂x2
0, or
∂2
φ
∂y2
0,









and
∂2
φ
∂x2
∂2
φ
∂y2
∂2
φ
∂x ∂y
2
Saddle point:
∂2
φ
∂x2
∂2
φ
∂y2
∂2
φ
∂x ∂y
2
If
∂2
φ
∂x2
=
∂2
φ
∂y2
=
∂2
φ
∂x ∂y
= 0 the character of the turning point is determined by the next higher derivative.
Changing variables: the chain rule
If φ = f (x, y, . . .) and the variables x, y, . . . are functions of independent variables u, v, . . . then
∂φ
∂u
=
∂φ
∂x
∂x
∂u
+
∂φ
∂y
∂y
∂u
+ · · ·
∂φ
∂v
=
∂φ
∂x
∂x
∂v
+
∂φ
∂y
∂y
∂v
+ · · ·
etc.
Page 16 of 24
Functions of Several Variables

18. Changing variables in surface and volume integrals – Jacobians
If an area A in the x, y plane maps into an area A0
in the u, v plane then
Z
A
f (x, y) dx dy =
Z
A0
f (u, v)J du dv where J =
∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
The Jacobian J is also written as
∂(x, y)
∂(u, v)
. The corresponding formula for volume integrals is
Z
V
f (x, y, z) dx dy dz =
Z
V0
f (u, v, w)J du dv dw where now J =
∂x
∂u
∂x
∂v
∂x
∂w
∂y
∂u
∂y
∂v
∂y
∂w
∂z
∂u
∂z
∂v
∂z
∂w
Fourier series
If y(x) is a function defined in the range −π ≤ x ≤ π then
y(x) ≈ c0 +
M
∑
m=1
cm cos mx +
M0
∑
m=1
sm sin mx
where the coefficients are
c0 =
1
2π
Z π
−π
y(x) dx
cm =
1
π
Z π
−π
y(x) cos mx dx (m = 1, . . . , M)
sm =
1
π
Z π
−π
y(x) sin mx dx (m = 1, . . . , M0
)
with convergence to y(x) as M, M0
→ ∞ for all points where y(x) is continuous.
Fourier series for other ranges
Variable t, range 0 ≤ t ≤ T, (i.e., a periodic function of time with period T, frequency ω = 2π/T).
y(t) ≈ c0 + ∑cm cos mωt + ∑sm sin mωt
where
c0 =
ω
2π
Z T
0
y(t) dt, cm =
ω
π
Z T
0
y(t) cos mωt dt, sm =
ω
π
Z T
0
y(t) sin mωt dt.
Variable x, range 0 ≤ x ≤ L,
y(x) ≈ c0 + ∑cm cos
2mπx
L
+ ∑sm sin
2mπx
L
where
c0 =
1
L
Z L
0
y(x) dx, cm =
2
L
Z L
0
y(x) cos
2mπx
L
dx, sm =
2
L
Z L
0
y(x) sin
2mπx
L
dx.
Page 17 of 24
Fourier Series and Transforms

19. Fourier series for odd and even functions
If y(x) is an odd (anti-symmetric) function [i.e., y(−x) = −y(x)] defined in the range −π ≤ x ≤ π, then only
sines are required in the Fourier series and sm =
2
π
Z π
0
y(x) sin mx dx. If, in addition, y(x) is symmetric about
x = π/2, then the coefficients sm are given by sm = 0 (for m even), sm =
4
π
Z π/2
0
y(x) sin mx dx (for m odd). If
y(x) is an even (symmetric) function [i.e., y(−x) = y(x)] defined in the range −π ≤ x ≤ π, then only constant
and cosine terms are required in the Fourier series and c0 =
1
π
Z π
0
y(x) dx, cm =
2
π
Z π
0
y(x) cos mx dx. If, in
addition, y(x) is anti-symmetric about x =
π
2
, then c0 = 0 and the coefficients cm are given by cm = 0 (for m even),
cm =
4
π
Z π/2
0
y(x) cos mx dx (for m odd).
[These results also apply to Fourier series with more general ranges provided appropriate changes are made to the
limits of integration.]
Complex form of Fourier series
If y(x) is a function defined in the range −π ≤ x ≤ π then
y(x) ≈
M
∑
−M
Cm eimx
, Cm =
1
2π
Z π
−π
y(x) e−imx
dx
with m taking all integer values in the range ±M. This approximation converges to y(x) as M → ∞ under the same
conditions as the real form.
For other ranges the formulae are:
Variable t, range 0 ≤ t ≤ T, frequency ω = 2π/T,
y(t) =
∞
∑
−∞
Cm eimωt
, Cm =
ω
2π
Z T
0
y(t) e−imωt
dt.
Variable x0
, range 0 ≤ x0
≤ L,
y(x0
) =
∞
∑
−∞
Cm ei2mπx0/L
, Cm =
1
L
Z L
0
y(x0
) e−i2mπx0/L
dx0
.
Discrete Fourier series
If y(x) is a function defined in the range −π ≤ x ≤ π which is sampled in the 2N equally spaced points xn =
nx/N [n = −(N − 1) . . . N], then
y(xn) = c0 + c1 cos xn + c2 cos 2xn + · · · + cN−1 cos(N − 1)xn + cN cos Nxn
+ s1 sin xn + s2 sin 2xn + · · · + sN−1 sin(N − 1)xn + sN sin Nxn
where the coefficients are
c0 =
1
2N ∑ y(xn)
cm =
1
N ∑ y(xn) cos mxn (m = 1, . . . , N − 1)
cN =
1
2N ∑ y(xn) cos Nxn
sm =
1
N ∑ y(xn) sin mxn (m = 1, . . . , N − 1)
sN =
1
2N ∑ y(xn) sin Nxn
each summation being over the 2N sampling points xn.
Page 18 of 24

20. Fourier transforms
If y(x) is a function defined in the range −∞ ≤ x ≤ ∞ then the Fourier transform b
y(ω) is defined by the equations
y(t) =
1
2π
Z ∞
−∞
b
y(ω) eiωt
dω, b
y(ω) =
Z ∞
−∞
y(t) e−iωt
dt.
If ω is replaced by 2πf, where f is the frequency, this relationship becomes
y(t) =
Z ∞
−∞
b
y( f ) ei2π f t
d f , b
y( f ) =
Z ∞
−∞
y(t) e−i2π f t
dt.
If y(t) is symmetric about t = 0 then
y(t) =
1
π
Z ∞
0
b
y(ω) cosωt dω, b
y(ω) = 2
Z ∞
0
y(t) cosωt dt.
If y(t) is anti-symmetric about t = 0 then
y(t) =
1
π
Z ∞
0
b
y(ω) sinωt dω, b
y(ω) = 2
Z ∞
0
y(t) sinωt dt.
Specific cases
y(t) = a, |t| ≤ τ
= 0, |t| τ
(‘Top Hat’), b
y(ω) = 2a
sinωτ
ω
≡ 2aτ sinc(ωτ)
where sinc(x) =
sin(x)
x
y(t) = a(1 − |t|/τ), |t| ≤ τ
= 0, |t| τ
(‘Saw-tooth’), b
y(ω) =
2a
ω2
τ
(1 − cosωτ) = aτ sinc2
ωτ
2
y(t) = exp(−t2
/t2
0) (Gaussian), b
y(ω) = t0
√
π exp −ω2
t2
0/4
y(t) = f (t) eiω0t
(modulated function), b
y(ω) = b
f (ω − ω0)
y(t) =
∞
∑
m=−∞
δ(t − mτ) (sampling function) b
y(ω) =
∞
∑
n=−∞
δ(ω − 2πn/τ)
Page 19 of 24

21. Convolution theorem
If z(t) =
Z ∞
−∞
x(τ)y(t − τ) dτ =
Z ∞
−∞
x(t − τ)y(τ) dτ ≡ x(t) ∗ y(t) then b
z(ω) = b
x(ω) b
y(ω).
Conversely, c
xy = b
x ∗ b
y.
Parseval’s theorem
Z ∞
−∞
y∗
(t) y(t) dt =
1
2π
Z ∞
−∞
b
y∗
(ω) b
y(ω) dω (if b
y is normalised as on page 21)
Fourier transforms in two dimensions
b
V(k) =
Z
V(r) e−ik·r
d2
r
=
Z ∞
0
2πrV(r)J0(kr) dr if azimuthally symmetric
Fourier transforms in three dimensions
Examples
V(r) b
V(k)
1
4πr
1
k2
e−λr
4πr
1
k2
+ λ2
∇V(r) ikb
V(k)
∇2
V(r) −k2 b
V(k)
b
V(k) =
Z
V(r) e−ik·r
d3
r
=
4π
k
Z ∞
0
V(r) r sin kr dr if spherically symmetric
V(r) =
1
(2π)3
Z
b
V(k) eik·r
d3
k
Page 20 of 24

22. If y(t) is a function defined for t ≥ 0, the Laplace transform y(s) is defined by the equation
y(s) = L{y(t)} =
Z ∞
0
e−st
y(t) dt
Function y(t) (t 0) Transform y(s)
δ(t) 1 Delta function
θ(t)
1
s
Unit step function
tn n!
sn+1
t
1/
2
1
2
r
π
s3
t−1
/
2
r
π
s
e−at 1
(s + a)
sinωt
ω
(s2 + ω2
cosωt
s
(s2 + ω2)
sinh ωt
ω
(s2 − ω2)
coshωt
s
(s2 − ω2)
e−at
y(t) y(s + a)
y(t − τ) θ(t − τ) e−sτ
y(s)
ty(t) −
dy
ds
dy
dt
sy(s) − y(0)
dn
y
dtn
sn
y(s) − sn−1
y(0) − sn−2
dy
dt
0
· · · −
dn−1
y
dtn−1
0
Z t
0
y(τ) dτ
y(s)
s
Z t
0
x(τ) y(t − τ) dτ
Z t
0
x(t − τ) y(τ) dτ









x(s) y(s) Convolution theorem
[Note that if y(t) = 0 for t 0 then the Fourier transform of y(t) is b
y(ω) = y(iω).]
Page 21 of 24
Laplace Transforms

23. Finding the zeros of equations
If the equation is y = f (x) and xn is an approximation to the root then either
xn+1 = xn −
f (xn)
f 0
(xn)
. (Newton)
or, xn+1 = xn −
xn − xn−1
f (xn) − f (xn−1)
f (xn) (Linear interpolation)
are, in general, better approximations.
Numerical integration of differential equations
If
dy
dx
= f (x, y) then
yn+1 = yn + h f (xn, yn) where h = xn+1 − xn (Euler method)
Putting y∗
n+1 = yn + h f (xn, yn) (improved Euler method)
then yn+1 = yn +
h[ f (xn, yn) + f (xn+1, y∗
n+1)]
2
Central difference notation
If y(x) is tabulated at equal intervals of x, where h is the interval, then δyn+1/2 = yn+1 − yn and
δ2
yn = δyn+1/2 − δyn−1/2
Approximating to derivatives
dy
dx
n
≈
yn+1 − yn
h
≈
yn − yn−1
h
≈
δyn+1
/
2
+ δyn−1
/
2
2h
where h = xn+1 − xn
d2
y
dx2
n
≈
yn+1 − 2yn + yn−1
h2
=
δ2
yn
h2
Interpolation: Everett’s formula
y(x) = y(x0 + θh) ≈ θy0 + θy1 +
1
3!
θ(θ
2
− 1)δ2
y0 +
1
3!
θ(θ2
− 1)δ2
y1 + · · ·
where θ is the fraction of the interval h (= xn+1 − xn) between the sampling points and θ = 1 − θ. The first two
terms represent linear interpolation.
Numerical evaluation of definite integrals
Trapezoidal rule
The interval of integration is divided into n equal sub-intervals, each of width h; then
Z b
a
f (x) dx ≈ h
c
1
2
f (a) + f (x1) + · · · + f (xj) + · · · +
1
2
f (b)
where h = (b − a)/n and xj = a + jh.
Simpson’s rule
The interval of integration is divided into an even number (say 2n) of equal sub-intervals, each of width h =
(b − a)/2n; then
Z b
a
f (x) dx ≈
h
3
f (a) + 4 f (x1) + 2 f (x2) + 4 f (x3) + · · · + 2 f (x2n−2) + 4 f (x2n−1) + f (b)
Page 22 of 24
Numerical Analysis

24. Gauss’s integration formulae
These have the general form
Z 1
−1
y(x) dx ≈
n
∑
1
ci y(xi)
For n = 2 : xi = ±0·5773; ci = 1, 1 (exact for any cubic).
For n = 3 : xi = −0·7746, 0·0, 0·7746; ci = 0·555, 0·888, 0·555 (exact for any quintic).
Sample mean x =
1
n
(x1 + x2 + · · · xn)
Residual: d = x − x
Standard deviation of sample: s =
1
√
n
(d2
1 + d2
2 + · · · d2
n)1/2
Standard deviation of distribution: σ ≈
1
√
n − 1
(d2
1 + d2
2 + · · · d2
n)1/2
Standard deviation of mean: σm =
σ
√
n
=
1
q
n(n − 1)
(d2
1 + d2
2 + · · · d2
n)1/2
=
1
q
n(n − 1)
∑x2
i −
1
n ∑xi
2
1/2
Result of n measurements is quoted as x ± σm.
Range method
A quick but crude method of estimating σ is to find the range r of a set of n readings, i.e., the difference between
the largest and smallest values, then
σ ≈
r
√
n
.
This is usually adequate for n less than about 12.
Combination of errors
If Z = Z(A, B, . . .) (with A, B, etc. independent) then
(σZ)2
=
∂Z
∂A
σA
2
+
∂Z
∂B
σB
2
+ · · ·
So if
(i) Z = A ± B ± C, (σZ)2
= (σA)2
+ (σB)2
+ (σC)2
(ii) Z = AB or A/B,
σZ
Z
2
=
σA
A
2
+
σB
B
2
(iii) Z = Am
,
σZ
Z
= m
σA
A
(iv) Z = ln A, σZ =
σA
A
(v) Z = exp A,
σZ
Z
= σA
Page 23 of 24
Treatment of Random Errors

25. Mean and Variance
A random variable X has a distribution over some subset x of the real numbers. When the distribution of X is
discrete, the probability that X = xi is Pi. When the distribution is continuous, the probability that X lies in an
interval δx is f (x)δx, where f (x) is the probability density function.
Mean µ = E(X) = ∑Pixi or
Z
x f (x) dx.
Variance σ2
= V(X) = E[(X − µ)2
] = ∑Pi(xi − µ)2
or
Z
(x − µ)2
f (x) dx.
Probability distributions
Error function: erf(x) =
2
√
π
Z x
0
e−y2
dy
Binomial: f (x) =
n
x
px
qn−x
where q = (1 − p), µ = np, σ2
= npq, p 1.
Poisson: f (x) =
µx
x!
e−µ
, and σ2
= µ
Normal: f (x) =
1
σ
√
2π
exp
−
(x − µ)2
2σ2
Weighted sums of random variables
If W = aX + bY then E(W) = aE(X) + bE(Y). If X and Y are independent then V(W) = a2
V(X) + b2
V(Y).
Statistics of a data sample x1, . . . , xn
Sample mean x =
1
n ∑xi
Sample variance s2
=
1
n ∑(xi − x)2
=
1
n ∑x2
i
− x2
= E(x2
) − [E(x)]2
Regression (least squares fitting)
To fit a straight line by least squares to n pairs of points (xi, yi), model the observations by yi = α + β(xi − x) + i,
where the i are independent samples of a random variable with zero mean and variance σ2
.
Sample statistics: s2
x =
1
n ∑(xi − x)2
, s2
y =
1
n ∑(yi − y)2
, s2
xy =
1
n ∑(xi − x)(yi − y).
Estimators: b
α = y, b
β =
s2
xy
s2
x
; E(Y at x) = b
α + b
β(x − x); b
σ2
=
n
n − 2
(residual variance),
where residual variance =
1
n ∑{yi − b
α − b
β(xi − x)}2
= s2
y −
s4
xy
s2
x
.
Estimates for the variances of b
α and b
β are
b
σ2
n
and
b
σ2
ns2
x
.
Correlation coefficient: b
ρ = r =
s2
xy
sxsy
.
Page 24 of 24
Statistics